A contractor combined x tons of a gravel mixture

[StaceyKoprince]

nice work guys!

[prajeen_v]

you can write the problem statement as 10%x +2%y=5%z

if you are given values of y and z, it can be solved for x.

so both statements together are sufficient. hence D.

yeah, but that's not what (d) means. (d) means that each of the statements, individually and alone, is sufficient to solve the problem.

here's the easiest way to do this:
FACT ABOUT WEIGHTED AVERAGES: if you have the weighted average and both endpoints, then you also have the RATIO of the weights in the problem. that ratio is the reciprocal of the ratio of the distances between the endpoints and the weighted average.
in this problem, we have this:
(endpoint Y 2%)-------distance=3-------(weighted average 5%)----------distance=5----------(endpoint X 10%)
so
since distance Y : distance X = 3 : 5, the ratio of the weights (literal "weights" in tons, in this problem) of Y : X must be 5 : 3.

because you have this ratio, specifying even one of the quantities is sufficient to determine everything - just use the ratio to figure out the rest.
therefore, either of the choices will be sufficient.

Hello Ron,
Thanks for your explanation.
I understood how the statement (1) is sufficient but i could not relate the ratio short cut and statement(2).
Can you please explain how we can prove the statement(2) to be sufficient from here on.
thanks

[RonPurewal]

Hello Ron,
...
Can you please explain how we can prove the statement(2) to be sufficient from here on.
thanks

sure

the key to proving statement 2 sufficient (without using the number line shortcut) lies in realizing that you don't need the two variables 'x' and 'y' anymore.

if you have statement 2, then you know that x and y must add up to 16 (since 'z' is the total number of tons when x and y are combined).
therefore, take out 'y' and just call it '16 - x'.

in fact, this is probably the single most important strategy in all of word translations:
NEVER use more variables than you absolutely NEED to use.
in particular, if two or more quantities are connected by some sort of simple relationship, then you generally shouldn't need to designate a separate variable for each quantity.

--

once you've done that, it's a pretty standard setup:
0.10(x) + 0.02(16 - x) = 0.05(16)

this is an equation that you can solve for x; since this is a data sufficiency problem, you don't actually have to solve the equation. (on the other hand, it's good practice to carry out the steps at least to the point where you can be sure that x won't cancel out of the equation!)

[prajeen_v]

sure

the key to proving statement 2 sufficient (without using the number line shortcut) lies in realizing that you don't need the two variables 'x' and 'y' anymore.

Hello Ron,
Thanks for your explanation. Can you also show some light on how to get X using ratio method when only statement(2) is considered. I understood the solution for the statement(1) but i am confused with statement(2).
thanks

[ChrisB]

Hi,

Good question. A few have discussed this already, but statement (2) provides the value of z. From the question text we know that z = x + y. From there then, we can still apply the same methodology Ron applied to tackle statement 1.

Please let us know whether that clears things up for you!

Best regards,
Chris

[Enrique17]

Hi Ron, Stacey,

Can we calculate like this:

10x+2y=5Z or 5(x+y)

1. y=10 sufficient to know x
2. z=(x+y)=16 sufficient to know x

Therefore, D.

yes, that's an excellent way to do it.

BUT
DO NOT, EVER, LEAVE THE SAME VARIABLE ON BOTH SIDES OF A LINEAR EQUATION.

in this case, this means that you should simplify 10x + 2y = 5(x + y) to 10x + 2y = 5x + 5y, which in turn simplifies to 5x = 3y.
in that case, it's a lot easier to see why the statements are sufficient; they're both easy substitutions at this point.

Hi Stacey, I seem to be stuck on how to see sufficiency in statement (2) z=16 (or x+y=16) to solve 5x=3y, which is the solution I had come to and wrongly chose Answer Choice A (for sufficiency only for Stmt 1).

[RonPurewal]

if you have 5x = 3y AND x + y = 16, then x and y can be found with ordinary high-school algebra techniques-- either substitution or elimination.

if this is not absolutely (and immediately) clear, then you should search the internet for practice materials ('systems of equations with 2 unknowns'), and practice until you've built a better intuition for / knowledge of this sort of thing.

[Enrique17]

Got it, Thanks Ron! A bit scary to see that not until explicitly being pointed out that substitution or elimination would have resulted in sufficiency for either statement. Will take up the advice.

[RonPurewal]

Got it, Thanks Ron! A bit scary to see that not until explicitly being pointed out that substitution or elimination would have resulted in sufficiency for either statement. Will take up the advice.

you're welcome.

i don't understand what the middle sentence says. (it's not a sentence; it seems to be missing some words.) but i think i get what you're saying.